Conformally flat noncircular spacetimes

Conformally flat noncircular spacetimes

Eloy Ayon-Beato,1,2,* Cuauhtemoc Campuzano,2,3,† and Alberto A. Garcıa2,4,‡

1Centro de Estudios Cientıficos (CECS), Casilla 1469, Valdivia, Chile2Departamento de Fısica, Centro de Investigacion y de Estudios Avanzados del IPN, Apdo. Postal 14-740, 07000, Mexico D.F., Mexico

3Instituto de Fısica, Pontificia Universidad Catolica de Valparaıso, Casilla 4950, Valparaıso4Department of Physics, University of California, Davis, California 95616, USA

(Received 6 April 2006; published 13 July 2006)

The general metric for conformally flat stationary cyclic symmetric noncircular spacetimes is explicitlyderived. In spite of the complexity (due to the noncircular contributions) of the partial differentialequations arising from the conformal flatness constraints, they allow for their complete integration interms of elementary functions. A canonical coordinate representation of the search metric structure isgiven. Conditions for the existence of a rotation axis (axisymmetry) are established; these conditionscoincide with those which restrict the studied noncircular class of spacetimes to be static. As aconsequence, a previously known theorem by Collinson is just a part of a more general result: anyconformally flat stationary cyclic symmetric spacetime (even a noncircular one) is additionally axisym-metric if and only if it is also static. Recent astrophysical motivations point in the direction of consideringstationary noncircular configurations to describe magnetized neutron stars.

DOI: 10.1103/PhysRevD.74.024014 PACS numbers: 04.20.Jb

I. INTRODUCTION

One of the questions still open in general relativity is thedetermination of enough general interior configurationsdescribing isolated rotating bodies supporting their corre-sponding exterior gravitational fields. Usually, the descrip-tion of these rotating configurations is done by means ofcircular spacetimes, i.e., stationary axisymmetric space-times where the metric, in addition to being time androtation-angle independent, also possesses as isometriesthe inversions of time and of the rotation angle. It isworthwhile to mention that almost all the interior station-ary axisymmetric configurations reported in the literaturebelong to this class, and most of them suffer from somedeficiency.

Nevertheless, there is no room for the circular ideal-ization when considering rotating neutron stars [1] sur-rounded by strong toroidal magnetic fields ranging from�1016 to 1017 G; see also [2,3] and references therein. Thecircularity condition is a very severe restriction, which failsto hold in spacetimes allowing the existence of toroidalmagnetic fields and meridional flows [3]. Thus, to dealwith such astrophysical configurations one has to abandonthe fulfillment of the circularity condition and consider inconsequence the wider noncircular class of spacetimes.

Besides the above astrophysically motivated reasons tostudy noncircular configurations, there are also purelytheoretical ones. As soon as Schwarzschild published hisexterior spherically symmetric static solution, he was ableto determine its interior solution modeled through a perfectfluid with homogeneous density. Later, in light of the

Petrov classification, it was established that theSchwarzschild solution belongs to Petrov type D, whilethe interior Schwarzschild solution falls in the conformallyflat family. In 1973 Kerr reported his famous stationaryaxisymmetric gravitational field corresponding to a fieldcreated by a rotating body; this solution also belongs toPetrov type D. The search for the interior solution to theexterior Kerr metric began after that time. Collinson estab-lished that a conformally flat stationary axisymmetricspacetime is necessarily static [4]. Some stationary axi-symmetric Petrov-D metrics (coupled to perfect fluid dis-tributions) have been reported in the literature, but none ofthem allows for the matching with the Kerr metric.Recently, the results of Ref. [5] indicate that the matchingof an interior noncircular spacetime with an exterior cir-cular one is at least technically possible. This fact opens upthe possibility of searching for interior solutions within thenoncircular class.

Moreover, the general metric for conformally flat sta-tionary cyclic symmetric circular metric has been reportedrecently [6]; see, in this respect, also [7,8]. For this metric,being cyclic but not axisymmetric, because of the lack of arotation axis, the circularity theorem [9] does not hold. Thenext step in complexity, which is the main goal of thepresent work, is to determine the metric for conformallyflat stationary cyclic symmetric noncircular spacetimes. Inparticular, from this more general result one is able toderive the particular circular branch, and if one requiresthe existence of an axis of symmetry one recovers thestaticity property of the considered class of metrics, thusthe Collinson theorem is just included within a moregeneral result.

In the next section the mathematical preliminariesneeded in order to study the spacetimes under considera-tion are introduced. Specifically, the physical and geomet-

*Electronic address: ayon-at-cecs.cl†Electronic address: cuauhtemoc.campuzano-at-ucv.cl‡Electronic address: aagarcia-at-fis.cinvestav.mx

PHYSICAL REVIEW D 74, 024014 (2006)

1550-7998=2006=74(2)=024014(9) 024014-1 © 2006 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.74.024014

rical details behind the concepts of stationarity, cyclicsymmetry, axisymmetry, circularity, and staticity areclearly stated in order to make the work self-contained.In Sec. III the conformal flatness conditions, consisting inthe vanishing of the complex Weyl components, are fullyintegrated for any stationary cyclic symmetric noncircularspacetime. Section IV is devoted to revising the conditionsguaranteeing staticity of the obtained spacetimes in theircanonical representation. In Sec. V the conditions for theexistence of the axis of symmetry for the derived space-times are analyzed. It is concluded that both properties—the staticity and existence of a rotation axis—restrict themetric to become circular and static; these conclusions arestated in Sec. VII.

II. STATIONARY CYCLIC SYMMETRICSPACETIMES

In this section we characterize stationary cyclic sym-metric spacetimes; see, for example, Ref. [10] for theoriginal definitions. A spacetime is stationary if it admitsan asymptotically timelike Killing field. A spacetime iscalled cyclic symmetric if it is invariant under the action ofthe one-parameter group SO�2�; it is assumed that thecorresponding Killing field m with closed integral curvesis spacelike. A cyclic symmetric spacetime is named axi-symmetric if the fixed point set of the SO�2� action, i.e., therotation axis, is nonempty. A spacetime is called stationarycyclic symmetric (axisymmetric) if it is both stationary andcyclic symmetric (axisymmetric) and if the Killing fields kand m commute.

A stationary cyclic symmetric (axisymmetric) spacetimeis said to be circular if the 2-dimensional surfaces orthogo-nal to the Killing fields k and m are integrable. This isequivalent to satisfying the Frobenius integrability condi-tions

m ^ k ^ dk � 0; k ^m ^ dm � 0: (1)

The circularity property means that locally the gravita-tional field is not only independent of time and the rotationangle, but, it is also invariant under the simultaneousinversion of time and the angle. Almost all the literaturerelated to stationary cyclic symmetric (axisymmetric)spacetimes concerns only the circular case. This is, inpart, for simplicity, since in this case it is possible to usethe Lewis-Papapetrou ansatz for the metric.

As a last definition, a stationary spacetime is said to bestatic if the Killing field k is hypersurface orthogonal. Thisoccurs if and only if it satisfies

k ^ dk � 0; (2)

and it is equivalent to demanding that locally the gravita-tional field is not only time independent but it is alsoinvariant under time reversal.

In this work we are interested in noncircular spacetimes,i.e., general stationary cyclic symmetric spacetimes not

necessarily restricted to satisfying the Frobenius integra-bility conditions (1). The metric of such spacetimes can bewritten as

g � e�2Q�� 1

a�b �d�� ad�� Im�Mdz��d�� bd��

Im�Ndz�� e�2Pdzd�z�; (3)

where a, b, P, and Q are real functions and M and N arecomplex ones. Here the bar means complex conjugation,and Im (respectively, Re) denotes the imaginary (respec-tively, real) part of a complex quantity. Because the Killingfields realizing the stationary and cyclic isometries are k �@� and m � @�, all functions appearing above depend onthe coordinates z and �z only. The above metric has eightindependent real functions; hence, the diffeomorphisminvariance still allows for two remaining gauge choices.Notice that this metric (3) is invariant under the coordinatetransformations

��; �; z; �z� !��; �;

Z��z�dz;

Z�� z�d�z

�; (4)

together with the redefinitions

P! P� ln��z� �� z�

q; M !

M��z�

; N !N��z�

:

(5)

This gauge of freedom will be exploited in the forthcomingsection to bring the final result into an optimal form.

The noncircularity property of the metric (3) becomesapparent from the nonvanishing of the following quanti-ties:

��m ^ k ^ dk� �e2�P�Q�

2�a� b�Re�@�M� N�

@�z

�M� Na� b

@�a� b�@�z

�; (6)

��k ^m ^ dm� �e2�P�Q�

2�a� b�Re�a@M@�z� b

@N@ �z

�M� Na� b

@�ab�@�z

�; (7)

where the star stands for the Hodge dual operation.To evaluate the Weyl tensor components, it is more

convenient to use the Newman-Penrose formalism.Writing the metric (3) as

g � 2e1e2 � 2e3e4; (8)

with respect to the complex null tetrad basis

AYON-BEATO, CAMPUZANO, AND GARCIA PHYSICAL REVIEW D 74, 024014 (2006)

024014-2

e1 �1��2p e�Q�Pdz; (9a)

e2 �1��2p e�Q�Pd �z; (9b)

e3 �1��2p

e�Q��a� bp

�d�� bd��

Ndz� �Nd �z�2i

�; (9c)

e4 �1��2p

e�Q��a� bp

�d�� ad��

Mdz� �Md �z2i

�; (9d)

one evaluates, via well-known standard procedures, thecorresponding Weyl complex components:

�0 �2e2�Q�P�

a� b

�@2a

@z2 � 2@P@z

@a@z�

2

a� b

�@a@z

�2�;

��4 �2e2�Q�P�

a� b

�@2b

@z2 � 2@P@z

@b@z�

2

a� b

�@b@z

�2�;

2�1 � �ie2Q�P��a� bp

�@@z�e2PM� �

e2P

a� b�M� 3N �

@a@z

�;

2 ��3 �ie2Q�P��a� bp

�@@z�e2PN � �

e2P

a� b�N � 3M�

@b@z

�;

6�2 �2e2�Q�P�

�a� b�2

�2�a� b�2

@2P@z@�z

� 5@a@z

@b@ �z�@a@�z

@b@z

�

�4e2Q�4P

a� bMN ; (10)

where for further convenience the following auxiliaryfunctions have been introduced:

M :� Re�@M@ �z�M� Na� b

@a@ �z

�;

N :� Re�@N@�z�M� Na� b

@b@ �z

�:

(11)

Moreover, these functions can be used to describe thenoncircularity properties of the metric; from Eq. (6) oneobtains

��m ^ k ^ dk� �e2�P�Q�

2�a� b��M�N �;

��k ^m ^ dm� �e2�P�Q�

2�a� b�

�aM� bN

�1

2Re�M� Na� b

@@ �z�a� b�2

��: (12)

For vanishing functions M and N, which in turn yield Mand N equal to zero, one gets back the circular case.

III. SOLVING THE CONFORMAL FLATNESSCONSTRAINTS

In order to determine the general class of conformallyflat stationary cyclic symmetric metrics, we demand thevanishing of all the Weyl complex components �0 ��4 � �2 � �1 � �3 � 0.

The complex components �0 and �4 occur to be equalto the corresponding ones of the circular case (M � 0 �N) studied in Refs. [4,6]. Hence, one can follow the samestrategy applied in [6] to integrate the equations for �0 and�4; the vanishing of the combinations

�0� ��4�2�a�b�e2Q @@z

�e2P

�a�b�2@�a�b�@z

��0;

�0@b@z� ��4

@a@z�2�a�b�e2�Q�P� @

@z

�e4P

�a�b�2@a@z@b@z

��0

(13)

gives rise to the following first order partial differentialequations for the structural functions a�z� and b�z�:

@a@z�@b@z� �g� �z��a� b�2e�2P; (14)

@a@z

@b@z� �h��z� �g2��z��a� b�2e�4P; (15)

where g�z� and h�z� are integration functions.Because of the real character of the functions a, b, and

P, Eq. (14) implies

g�z�@@z�a� b� � �g� �z�

@@�z�a� b�: (16)

Using the freedom of the coordinate z, Eq. (4), togetherwith the redefinition of P from Eq. (5), by choosing � � g,without any loss of generality one can set g�z� � 1 inEq. (14), and consequently the same substitution can bedone in Eq. (16), which yields

a�z; �z� � b�z; �z� � F�z� �z�: (17)

Substituting the function b�z; �z� from Eq. (17) into theimaginary part of the component �2,

Im ��2� �e2�Q�P�

i�a� b�2

�@a@z

@b@ �z�@a@ �z

@b@z

�� 0; (18)

one arrives at

dFd�z� �z�

�@a@z�@a@�z

�� 0; (19)

therefore, if F � const then necessarily a � a�z� �z�, andb � b�z� �z�, and as a consequence of Eq. (14) one con-cludes that P � P�z� �z�. As far as the function �h��z�appearing in Eq. (15) is concerned, because of the depen-dence of a, b, and P on the variable z� �z, one easilyestablishes, using Eq. (15), that �h��z� � h�z� � const ��k2, where � � 1 just encodes the signs of the constant.The relevance of both signs of � was pointed out inRef. [6], and they were used to establish new results.

Introducing the real and imaginary parts of z � x� iyas coordinates, and the notations for the real and imaginaryparts of M�x; y� � MR�x; y� � iMI�x; y� and N�x; y� �NR�x; y� � iNI�x; y�, the metric (3) can be written as

CONFORMALLY FLAT NONCIRCULAR SPACETIMES PHYSICAL REVIEW D 74, 024014 (2006)

024014-3

g � e�2Q��

1

a� b�d�� ad�� MRdy�MIdx��d�

� bd�� NRdy� NIdx�� e�2P�dx2 � dy2�

�:

(20)

Now, we apply the same strategy of Ref. [6] to solveEqs. (14) and (15), which are now expressed as

dadx�dbdx� �a� b�2e�2P; (21)

dadx

dbdx� �k2�a� b�2e�4P: (22)

First we redefine the functions a and b through X and Y by

a� b � 2kY; a � k�Y � X�;

a� b � 2kX; b � k�Y � X�:(23)

Using these new functions X and Y, Eqs. (21) and (22)are rewritten correspondingly as

dXdx� 2kY2e�2P; (24)

�dYdx

�2�

�dXdx

�2� 4�k2Y2e�4P: (25)

Equation (24) suggests choosing a new coordinate x0 de-fined by

x0 � 2kZY2e�2Pdx: (26)

Consequently,

dx �1

2kY�2e2Pdx0;

ddx� 2kY2e�2P d

dx0: (27)

Thus Eqs. (24) and (25) become

dXdx0� 1; (28)

�dYdx0

�2�� Y2

Y2 : (29)

The general solutions of Eqs. (28) and (29) are X�x0� � x0

and Y�x0� ��x0 � x00�

2 � �q

. Without any loss of general-ity one can equate x00 to zero by replacing x0 ! x0 � x00 and�! �� x00. Hence,

X�x0� � x0; Y�x0� ��x02 � �

p: (30)

Returning to the complex conformal coefficients, from thevanishing of Re��1� and Re��3�,

Re�2 ��3� �ie2Q�3P

2��a� bp

�2N

�@@z�@@�z

�P�

�@@z�@@ �z

�N

�N � 3M

a� b

�@@z�@@�z

�b�;

Re�2�1� � �ie2Q�3P

2��a� bp

�2M

�@@z�@@ �z

�P

�

�@@z�@@�z

�M�

M� 3N

a� b

�@@z�@@ �z

�a�;

(31)

one concludes that

M �M�z� �z�; N �N �z� �z�: (32)

Introducing two new auxiliary functions

� �e2P

a� bM; � �

e2P

a� bN ; (33)

from the vanishing of the full expression of �1 and �3, oneobtains first order differential equations for the functions��z� �z� and ��z� �z�, correspondingly,

�a� b�@�@z� �

@b@z� 3�

@a@z� 0;

�a� b�@�@z� �

@a@z� 3�

@b@z� 0:

(34)

In terms of the variable x0 [Eq. (26)] and functions X�x0�and Y�x0� [Eq. (30)] one has

2Yd�dx0� �

ddx0�Y � X� � 3�

ddx0�Y � X� � 0;

2Yd�dx0� �

ddx0�Y � X� � 3�

ddx0�Y � X� � 0:

(35)

Explicitly,

2Y2 d�dx0� ��x0 � Y� � 3��x0 � Y� � 0;

2Y2 d�dx0� ��x0 � Y� � 3��x0 � Y� � 0:

(36)

Introducing a pair of new functions A�x0� and B�x0� through

��x0� � A�x0� � B�x0�; ��x0� � A�x0� � B�x0�; (37)

by adding and subtracting the above differential equations,one arrives at the simple system

Y2 ddx0

A�x0� � 2x0A�x0� � 2YB�x0� � 0; (38)

Y2 ddx0

B�x0� � x0B�x0� � YA�x0� � 0; (39)

which can be solved by using the method of increasing theorder of the differential equations; differentiating Eq. (39)one arrives at


024014-4

Y2 d2

dx02B� x0

ddx0

B� B� Yddx0

A�x0

YA � 0: (40)

Substituting into this equation the derivative ddx0 A from

Eq. (38) followed by the substitution of A�x0� fromEq. (39), one gets

�x02 � ��2d2

dx02B� x0�x02 � ��

ddx0

B� �B � 0; (41)

which allows for a further simplification by replacing B�x0�as B�x0� � V�x0�=

��x02 � �p

, namely

d2

dx02V�x0� � 0; ! V�x0� � C1x

0 � C0: (42)

Consequently, using Eq. (39) to evaluate A�x0�, one has

B�x0� �C1x0 � C0��x02 � �p ; A�x0� �

�x02 � ��C1 � 2x0C0

x02 � �:

(43)

From Eq. (37) one determines the search functions �and �, namely

��x0� ��x02 � �

x02 � ��

x0��x02 � �p

�C1

�

�2x0

x02 � ��

1��x02 � �p

�C0;

��x0� ��x02 � �

x02 � ��

x0��x02 � �p

�C1

�

�2x0

x02 � ��

1��x02 � �p

�C0: (44)

On the other hand, from the equation

Re�3�2� � 2e2Q�2P

a� b

��a� b�

@2P@z@�z

�1

a� b

�@a@z

@b@ �z�@a@ �z

@b@z

�� e2PMN

�� 0;

(45)

one establishes that

Y2 d2

dx02�Ye�2P� �

4

k��x0��x0� � 0: (46)

The integral of the above equation yields

e�2P�x0� �K1x0 � K0

Y�

4

kY

ZZ ��

Y2 dx0dx0: (47)

Replacing ��x0� and ��x0� from Eq. (44) one arrives at

e�2P�x0� �K1x0 � K0

Y�

2

kY2x0C0C1 � �C2

1 � C20

x02 � �; (48)

which, by replacing K0 ! K0 � 2C21=k, can be brought to

the form

e�2P�x0� �K1x0 � K0

Y�x0��

2

kY�x0�B�x0�2: (49)

Still it remains to integrate the first order equations con-tained in M and N for the real functionsMR,NR,MI, andNI, which explicitly amount to

2M � 4kYe�2P��x0�

� 2kY2e�2P�x�@@x0

MR �1

2Y

�x0

Y� 1

��MR � NR�

�

�@@yMI;

2N � 4kYe�2P��x0�

� 2kY2e�2P�@@x0

NR �1

2Y

�x0

Y� 1

��MR � NR�

�

�@@yNI: (50)

At this stage it will be useful to introduce the definitions

~N I :� e2P�x0� NI2kY

; ~MI :� e2P�x0� MI

2kY: (51)

Substituting in the metric (20) the primed coordinate x0,dx0 � 2kY2e�2Pdx, and using Eq. (51), dropping theprimes in what follows , the general conformally flat metricis rewritten as

g � e�2Q��

1

a� b�d�� ad�� MRdy� ~MIdx��

�d�� bd�� NRdy� ~NIdx��

�1

4k2 Y�4e2Pdx2 � e�2Pdy2

�: (52)

The remaining equations for the unknown functionsMR�x; y�, NR�x; y� and ~NI�x; y�, ~NI�x; y� are given by

@@xMR �

1

2Y

�xY� 1

��MR � NR� �

@@y

~MI

�2

Y�A�x� � B�x��;

@@xNR �

1

2Y

�xY� 1

��MR � NR� �

@@y

~MI

�2

Y�A�x� � B�x��:

(53)

Thus, there are various possibilities for searching for thesolutions of this system.

A. Nondiagonal �xy�-metric sector

Let us start by considering the functions MR�x; y� andNR�x; y� as the ones to be determined. Introducing thefunctions


024014-5

2F��x; y� :� MR � NR; 2F��x; y� :� MR � NR;

(54)

and adding Eqs. (53), one obtains

�x2 � ��@@x

F��x2 � �p �

�2A�x� �

@@y

~MI �@@y

~NI

�; (55)

with the general solution

F��x; y� � �2B�x� ��x2� �

pf��y�

��x2 � �

p Z 1

x2� �

�@@y

~MI �@@y

~NI

�dx: (56)

Next, subtracting Eqs. (53) one arrives at

��x2 � �

p @@xF� � 2B�x� � F� �

@@y

~MI �@@y

~NI: (57)

Using the expression of F� from Eq. (56) and integrating,one gets

F��x; y� � f��y�x�Z 1��

x2 � �p

�@@y

~MI �@@y

~NI

�dx

�ZZ 1

x2 � �

�@@y

~MI �@@y

~NI

�dxdx: (58)

Notice that the function f��y�, entering inMR as f��y�a�x�and in NR as �f��y�b�x�, can be eliminated from themetric by a shifting of the coordinate �, d�!d�� f��y�dy. Therefore the structural functions MR andNR are

MR�x; y� � �2B�x� � YZ 1

Y

�@@y

~MI �@@y

~NI

�dx

�Z 1

Y

�@@y

~MI �@@y

~NI�dx

�ZZ 1

Y2

�@@y

~MI �@@y

~NI

�dxdx; (59)

NR�x; y� � �2B�x� � YZ 1

Y

�@@y

~MI �@@y

~NI

�dx

�Z 1

Y

�@@y

~MI �@@y

~NI

�dx

�ZZ 1

Y2

�@@y

~MI �@@y

~NI

�dxdx (60)

which, together with the functions

Y�x� ��x2 � �

p; a�x� � k�Y � x�;

b�x� � k�Y � x�;

e�2P�x� �K1x� K0

Y�x��

2

kY�x�

�C1x� C0��x2 � �p

�2;

(61)

determine completely the conformally flat noncircularmetric

g � e�2Q��

1

a� b�d�� ad�� MRdy� ~MIdx��

�d�� bd�� NRdy� ~NIdx��

�1


�; (62)

where the functions ~MI�x; y� and ~NI�x; y� still remain free.A second possible representation can be achieved by

integrating Eq. (53) for the imaginary parts ~MI�x; y� and~NI�x; y� of M�x; y� and N�x; y�, respectively, arriving at

~MI�x; y� � �2��x�Y

y�m�x� �Z � @

@xMR

�1

2Y

�xY� 1

��MR � NR�

�dy;

~NI�x; y� � �2��x�Y

y� n�x� �Z � @

@xMR

�1

2Y

�xY� 1

��MR � NR�

�dy; (63)

where m�x� and n�x� are integration functions, while thefunctions MR�x0; y� and NR�x0; y� remain as free functions.Via an appropriate choice of new � and � coordinates, onecan set m�x� and n�x� to zero. Therefore, in this coordinategauge, the search metric structure can be represented bythe metric (62) with structural functions ��x� and ��x�given by unprimed Eqs. (44) and structural functions a�x�,b�x�, P�x�, and Y�x� given by Eq. (61), together with~MI�x; y� and ~MI�x; y� from Eqs. (63), while the structural

functions Q�x; y�, MR�x; y�, and NR�x; y� remain as freefunctions.

The noncircularity properties, guaranteed by the non-vanishing of Eqs. (12), can be written in terms of ��x� and��x�.

According to a general theorem of geometry, any 2-dimensional metric g�x; y� can be locally brought to thediagonal form, i.e., to its canonical representation, bymeans of coordinate transformations involving both xand y coordinates.

B. Canonical diagonal �xy� sector; gxy � 0

One might choose a canonical gauge of coordinates xand y for which the metric �xy� sector assumes a diagonalform, that is, with vanishing metric component gxy.

This representation can also be achieved from the abovegeneral metric expression Eq. (62) by equating ~MI�x; y�and ~NI�x; y� to zero. In this way one arrives at the canonicalexpression

g � e�2Q��

1

a� b�d�� ad��MRdy��d�� bd�

� NRdy� �1


�; (64)

with structural functions


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Y�x� ��x2 � �

p; B�x� �

C1x� C0��x2 � �p ;

a � k�Y � x�; b � k�Y � x�;

MR�x� � �2B�x�; NR�x� � �2B�x�;

e�2P�x� �K1x� K0

Y�x��

2

kY�x�B�x�2:

(65)

As a last step, in order to write the obtained metric in asimple form, we carry out the following coordinate trans-formation, ��; �; x; y� ! ��=

��2p; �=�

��2pk�; x; y=�2k��, to-

gether with the following redefinition of the conformalfactor, Q! Q� 1

2 ln�4k2Y�, and also simple redefinitionsof the involved constants, �C1; C0� ! �kC1=

��2p; kC0=

��2p�.

The general canonical form of the conformally flat sta-tionary cyclic symmetric metric can be given as

g � e�2Q�x;y��k�d�2 � 2xd�d�� d�2�

�dx2

�K0 � K1x��x2 � �� k�C0 � C1x�2

� 2k�C0 � C1x�d�dy� �K0 � K1x�dy2

�: (66)

It is easy to note that for C0 � 0 � C1 we recover thecircular metrics of Ref. [6]. Instead, forC0 � 0 andC1 � 0the above metric is noncircular as follows from the non-vanishing quantities,

� �m ^ k ^ dk� � k2e�2Q�C0 � C1x�;

��k ^m ^ dm� � k2e�2Q��C1 � C0x�:(67)

Since one can always achieve the canonical representationof the studied metric structures, it will be enough to estab-lish the properties of staticity and axisymmetry, if any, withrespect to the canonical metric (66).

IV. STATICITY

As it was defined in Sec. II, the spacetimes derived in theprevious section would be static if there exists a timelikelinear combination of the Killing fields,

ks � A0@@�� B0

@@�

; (68)

satisfying the staticity condition (2). For metric (66) such acondition becomes

0 � ��ks ^ dks�

� ke�2Q�x;y��kB0�xCb � Ca�d�� kA0�xCb � Ca�d�

� ��K1x� K0�Cc � kB0�C1x� C0�Cb�dy�;

Ca :� C0A0 � �C1B0; Cb :� C1A0 � C0B0;

Cc :� A20 � �B

20: (69)

It is straightforward to realize that we are in the presence of

static configurations only if Ca � 0 � Cb � Cc, whichyields

� � 1; A0 � B0; and C1 � C0: (70)

In such case, the hypersurface orthogonal Killing fields areproportional to ks � @=@� @=@�.

V. AXISYMMETRY

Now we turn our attention to the existence of a rotationaxis, i.e., if there are conformally flat stationary axisym-metric configurations within the class (66). It follows fromthe definition of Sec. II that the rotation axis is the space-time region where the cyclic Killing field m vanishes. Formetric (66), its general stationary and cyclic Killing fieldsare a linear combination of the vectors @� and @�. Hence,performing the transformation �� t� ��;� � �t��, where �� 0, the Killing fields are written ask � @t and m � @�, respectively. In terms of the newcoordinates, all the metric components g� � g�m; @�must vanish on the axis, which implies the following setof algebraic equations:

g�t � �ke�2Q�� x� �� 0; (71a)

g�� ke�2Q��2 � 2��x� ��2� � 0; (71b)

g�y � ke�2Q��C0 � C1x� � 0: (71c)

Isolating x from Eq. (71b) and inserting the result inEq. (71a), we obtain

�� 2 � ��2�

�� 0: (72)

Since �� 0 the above equation has nontrivialsolutions only if � � 1. Using the above conditions inthe remaining equation (71c), we obtain

��C1 C0� � 0; (73)

which implies that the related rotation axis must be locatedat

x � 1: (74)

Summarizing, metric (66) describes a stationary axisym-metric spacetime only for � � 1 and C1 � C0, i.e., theconditions for the existence of the rotation axis are thesame ones that guarantee the static character of the space-time; see conditions (70).

As a consequence, the previously known theorem byCollinson [4,6] is not only generalized to include configu-rations which are not necessarily circular, but, it is part of amore general statement: any conformally flat stationarycyclic symmetric spacetime (even a noncircular one) isadditionally axisymmetric if and only if it is also static.


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VI. CASE a � b; THE STATIC METRIC

This section deals with the remaining case, a� b �F � const. By shifting the timelike coordinate and at thesame time redefining the function a or b, the constant Fcan be equated to zero and hence this case is equivalent tohaving a � b.

From the list of the � complex components, Eqs. (10),we see that now �0 � ��4. Thus, from the vanishing of the�0 component one arrives at

�0 � e2Q @@z

�e2P @

@zlna

�� 0)

@a@z� �g��z�ae�2P; (75)

which, taking into account the real character of a and P,yields

1

�g��z�@a@z�

1

g�z�@a@�z: (76)

Again, just applying the coordinate transformation[Eqs. (4)] together with the relevant redefinitions of thefunctions P, M, and N [Eqs. (5)], by choosing � � g onecan set g�z� � 1. Therefore, by virtue of Eq. (76) thefunction a depends on z� �z only, a�z� �z�, and as aconsequence of Eq. (75) one concludes that P is a functionof z� �z too, or in terms of the real coordinates x � �z��z�=2 and y � �i�z� �z�=2, one has a � a�x� and P �P�x�. With all generality, we can rewrite Eq. (75) as

dadx� ae�2P ) dx �

daa

e2P; (77)

thus a can be considered as a new spatial coordinateinstead of x.

From the vanishing of the imaginary parts of �1 and �3

one arrives at

M �M�z� �z� �M�x�;

N �N �z� �z� �N �x�:(78)

In terms of the variable a, from their definitions through thestructural functions M�x; y� � MR�x; y� � iMI�x; y� andN�x; y� � NR�x; y� � iNI�x; y�, Eq. (11), one has

2M�a� � ae�2P @MR

@a�

1

2�MR � NR�e�2P �

@MI

@y

�: 4a��a�e�2P;

2N �a� � ae�2P @NR@a�

1

2�MR � NR�e

�2P �@NI@y

�: 4a��a�e�2P;

(79)

which are constrained to the equations arising from thevanishing of the full expression of �1 and �3, namely

2ad�dx� �� 3� � 0; 2a

d�dx� �� 3� � 0; (80)

which allow a straightforward integration,

��a� � C0a�2 � C1a; ��a� � C0a�2 � C1a; (81)

where C0 and C1 are constants of integration.The equation for the function P arising from the vanish-

ing of �2 reads

a4 d2

da2 e�2P � a3 dda

e�2P � a2e�2P � �16�C2

0

a� C2

1a5

�;

(82)

with the general solution

e�2P �K0

a� K1a� 2

C20

a3 � 2C21a

3: (83)

Finally, introducing the definitions

~M I�a; y� �e2P

aMI�a; y�; ~NI�a; y� �

e2P

aNI�a; y�;

(84)

the studied metric is rewritten as

g � e�2Q��

1

2a�d�� ad�� MRdy� ~MIda��

�d�� ad�� NRdy� ~NIda��

�1

a2 e2Pda2 � e�2Pdy2

�; (85)

and the integration of Eqs. (79) for MR and NR yields

MR�a; y� � �2a�� aF1�y� � F0�y�

�1

2

ZZ 1

a2

�@@y

~MI�a; y� �@@y

~NI�a; y��dada

�ZZ 1

a@2

@a@y~MI�a; y�dada;

NR�a; y� � �2a�� aF1�y� � F0�y�

�1

2

ZZ 1

a2

�@@y

~MI�a; y� �@@y

~NI�a; y��dada

�ZZ 1

a@2

@a@y~MI�a; y�dada: (86)

It becomes apparent that by shifting the � and � coordi-nates, d�� F0�y�dy! d� and d�� F0�y�dy! d�, onecan eliminate the functions F1�y� and F0�y�; thus, with allgenerality, one can set F1�y� � 0 � F0�y�.

Finally, scaling the Killing coordinates and constantsaccording to ��; �;C0; C1� !

��=��2p; �=

��2p; C0=

��2p; C1=

��2p�, the metric in canonical

�ay� coordinates (gay � 0) for a stationary cyclic symmet-ric spacetime with a � b amounts to


024014-8

g � e�2Q�x;y��d�2

a� ad�2

�1

a2

da2

K0=a� K1a� C20=a

3 � C21a

3

� 2�C1ad��

C0

ad�

�dy�

�K0

a� K1a

�dy2

�: (87)

Notice that the noncircular character of this metric isguaranteed by the nonvanishing of

� �m ^ k ^ dk� � �2aC1e�2Q;

��k ^m ^ dm� � 2C0

ae�2Q:

(88)

For C0 � 0 � C1 we recover the static metric of Ref. [6].The above metric describes a static spacetime for C1 � 0.Additionally, it allows for the existence of an axis ofrotation, hence this class is in agreement with Collinson’stheorem.

VII. CONCLUSIONS

In this paper we study all the stationary cyclic symmetricspacetimes which are at the same time conformally flat. Incontrast to previous works on the subject, we also considernoncircular configurations. The conformal flatness condi-tion is imposed on the metric structural functions by de-manding the vanishing of the Weyl tensor. The resultingconstraints are quite involved due to the inclusion of non-circular contributions. However, one can still accomplishthe complete integration of the system of ten nonlinearpartial differential equations, as has been shown in detail inSec. III. The class of obtained spacetimes is fully deter-mined up to an arbitrary conformal factor and it is endowedwith several constants. In particular, two constants charac-terize the noncircular behavior of these spacetimes; whenthey vanish, we recover the circular configurations ob-tained in Ref. [6].

The conditions allowing for the existence of a rotationaxis in the resulting configurations are also established.Moreover, the static branch of spacetimes is consideredtoo. The set of values of the constant is the same for bothphysical situations; the resulting spacetimes are axisym-metric if and only if they are also static. Hence, one of themain results of the paper can be summarized by the follow-ing statement.

Theorem—Any conformally flat stationary cyclic sym-metric spacetime (even a noncircular one) is additionallyaxisymmetric if and only if it is also static.

Within the properly cyclic symmetric class of metrics(with no rotation axis and, as a consequence of the abovetheorem, necessarily nonstatic), it should be of interest toinvestigate what kind of sources can solve Einstein equa-tions. In the case that one could retain the noncircularcontributions when matter is present, the derived configu-rations should be of particular interest for the description ofastrophysical relevant objects. Moreover, the present coor-dinate gauge and null tetrad choice could permit a success-ful treatment of other Petrov types because of the highlysymmetric representation of the Weyl complex compo-nents; a research program in this direction is in progress.

ACKNOWLEDGMENTS

We thank M. Hassaıne for discussions. This work hasbeen partially supported by FONDECYT GrantNo. 1040921, No. 7040190, and No. 1051064;CONACyT Grant No. 38495E, No. 48779F, andNo. 34222E; CONICYT/CONACyT Grant No. 2001-5-02-159; and MECESUP Grant No. FSM 0204. Partialsupport from CONACyT–MEXUS Grant No. 2005-0901and the assistance of S. Carlip are acknowledged byA. A. G. The generous support of Empresas CMPC to theCentro de Estudios Cientıficos (CECS) is also acknowl-edged. CECS is a Millennium Science Institute and isfunded in part by grants from Fundacion Andes and theTinker Foundation.

[1] E. Gourgoulhon and S. Bonazzola, Phys. Rev. D 48, 2635(1993).

[2] K. Ioka and M. Sasaki, Phys. Rev. D 67, 124026 (2003).[3] K. Ioka and M. Sasaki, Astrophys. J. 600, 296 (2004).[4] C. D. Collinson, Gen. Relativ. Gravit. 7, 419 (1976).[5] R. Vera, Classical Quantum Gravity 20, 2785 (2003).[6] A. A. Garcıa and C. Campuzano, Phys. Rev. D 66, 124018

(2002); 68, 049901(E) (2003).[7] A. Barnes and J. M. M. Senovilla, gr-qc/0305091.[8] A. A. Garcıa and C. Campuzano, gr-qc/0310054.[9] W. Kundt and M. Trumper, Z. Phys. 192, 419 (1966).

[10] M. Heusler, Black Hole Uniqueness Theorems (CambridgeUniversity Press, Cambridge, England, 1996).


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Conformally flat noncircular spacetimes